The WolframAlpha function gives to access to Wolfram|Alpha in Mathematica. Just like the Wolfram|Alpha web interface, there is a time limit on calculations. Also, you can't run arbitrary Mathematica programs. You need to use the "standard" W|A inputs.

While you can extend the time limit by subscribing to W|A Pro, I think what you are looking for is cloud computing services that support Mathematica. Here's something I managed to google up:

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